I was reading up on atheism when i came across Pascal's Wager at Stanford Encyclopedia of Philosophy. If you're not interested on going through the link, i'll give a brief introduction to Blaise Pascal's, seventeenth century mathematician, argument for believing in god (Catholic god, but i guess it applies to all christianity) in the form of a wager. The argument, known as Pascal's Wager, has since become one of the more famous argument in the philosophy of religion.

The wager is rather similar to Game Theory, except that it is a single person game. Pascal forms the wager that one could wager for god to exist or wager for god's non-existence. However a person wagers, the payoff would be determined by whether god exists anot.

Excerpt :
Suppose that you have two possible actions, A1 and A2, and the worst outcome associated with A1 is at least as good as the best outcome associated with A2; suppose also that in at least one state of the world, A1's outcome is strictly better than A2's. Let us say in that case that A1 superdominates A2. Then rationality surely requires you to perform A1.

Pascal's wager operates on three premise. The first premise is that of superdominance of one choice over the other. In simpler terms, it would become, "You've got nothing to lose in believing if god didn't exist and something to gain in believing if god does indeed exist.". The following is the table of payoffs.



However, we should note that:
Excerpt :
Without any assumption about your probability assignment to God's existence, the argument is invalid. Rationality does not require you to wager for God if you assign probability 0 to God existing.

Pascal then reduces the complexity of probability to either god exists or god don't (50%). The fallacy can be made clear by saying that 4D is 50% chance, win or lose. But we all know that (disregarding 2nd price and below) at four digits, the chance of winning is 0.01% rather than 50%.

Excerpt :
But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. It is all divided; wherever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all...

In the last premise, Pascal says that it is worth to gamble finite life for infinite prize, simply because of infinite rewards. As given in the following formula.



Excerpt :
However, Duff 1986 and Hájek 2003 argue that the argument is in fact invalid. Their point is that there are strategies besides wagering for God that also have infinite expectation — namely, mixed strategies, whereby you do not wager for or against God outright, but rather choose which of these actions to perform on the basis of the outcome of some chance device. Consider the mixed strategy: "Toss a fair coin: heads, you wager for God; tails, you wager against God". By Pascal's lights, with probability 1/2 your expectation will be infinite, and with probability 1/2 it will be finite. The expectation of the entire strategy is:



A rather funny take on the fallacy of the argument.
Excerpt :
That is, the ‘coin toss’ strategy has the same expectation as outright wagering for God. But the probability 1/2 was incidental to the result. Any mixed strategy that gives positive and finite probability to wagering for God will likewise have infinite expectation: "wager for God iff a fair die lands 6", "wager for God iff your lottery ticket wins", "wager for God iff a meteor quantum tunnels its way through the side of your house", and so on.

The problem is still worse than this, though, for there is a sense in which anything that you do might be regarded as a mixed strategy between wagering for God, and wagering against God, with suitable probability weights given to each. Suppose that you choose to ignore the Wager, and to go and have a hamburger instead. Still, you may well assign positive and finite probability to your winding up wagering for God nonetheless; and this probability multiplied by infinity again gives infinity. So ignoring the Wager and having a hamburger has the same expectation as outright wagering for God. Even worse, suppose that you focus all your energy into avoiding belief in God. Still, you may well assign positive and finite probability to your efforts failing, with the result that you wager for God nonetheless. In that case again, your expectation is infinite again. So even if rationality requires you to perform the act of maximum expected utility when there is one, here there isn't one.

And the moral implications of wagering for god?
Excerpt :
Clifford 1986 argues that an individual's believing something on insufficient evidence harms society by promoting credulity. Penelhum 1971 contends that the putative divine plan is itself immoral, condemning as it does honest non-believers to loss of eternal happiness, when such unbelief is in no way culpable; and that to adopt the relevant belief is to be complicit to this immoral plan.